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The instantaneous capital market line.

Article Excerpt
Summary. We show that if the intercept and slope of the instantaneous capital market line are deterministic, then investors will not hold any hedge portfolios in the sense of Merton [9, 11]. They will choose portfolios that plot on the capital market line, and they will slide up and down the capital market line over time as their wealth and risk tolerance change. This result allows us to aggregate over investors and derive a single factor CAPM where the first and second moments of security returns may change stochastic cally over time and markets are potentially incomplete.

Keywords and Phrases: Portfolio optimization, Incomplete markets, Capital market line, Mutual fund separation.

JEL Classification Numbers: G11, G12.

1 Introduction

This paper shows that if the intercept and slope of what we call the instantaneous capital market line (ICML) are deterministic, then investors will not hedge against changes in the means, variances or covariances of security returns.

We formulate our result in the form of a two-fund separation theorem. It says that if the ICML is deterministic, then investors will simply hold a possibly time-varying combination of two funds that span the ICML: the riskless asset and the logarithmic portfolio. Investors will place themselves along the ICML, and they will slide up and down the ICML over time as their wealth and risk tolerance change.

The theorem holds in a general framework with securities whose rates of return have stochastically time-varying first and second moments which do not have to be functions of a Markovian vector of state variables. The investor's utility is a state-independent function of his consumption over time and of his final wealth at the horizon time. Markets are not assumed to be dynamically complete.

The key to the proof is to create a simplified securities market model with only two assets, the money market account and the logarithmic portfolio, and then show that risk averse investors will prefer payoffs that can be replicated within this model.

The capital market line (CML) has always played a central role in static mean-variance portfolio theory. It would be graphed and used to visualize the determination of the optimal portfolio. Merton [8] showed that if the interest rate and all the first and second moments are constant, then the ICML has the same role in continuous time as the CML has in a static model. Our result implies that the ICML plays this same role in a wider set of circumstances.

The results in the literature that are closest to ours are found in Karatzas, Lehoczky, Shreve, and Xu [5] and Ocone and Karatzas [14]. Both assume complete markets, and Karatzas et al. [5] also assume a power utility function. The mathematical methods they use are quite different from ours: Ocone and Karatzas use Malliavin calculus, whereas Karatzas et al. do an explicit calculation relying on the parameter of the utility function. Both find that if the interest rate and the entire vector of prices of risk (as opposed to the length of this vector, which is the slope of the ICML) are deterministic, then investors do not need to hedge. This result cannot be recast in terms of the slope of the ICML and thus cannot be used to establish the ICML as the investment opportunity set.

Our result is similar in spirit to those of Constantinides [3]. He identifies some circumstances in which investors will not hold hedge portfolios even though some asset returns may be non-stationary, and in particular, even though some assets may have stochastically time-varying means, variances, and correlations. Specifically, he shows that this is true in equilibrium if the investors' utility functions have the aggregation property and all assets in positive supply have stationary returns. Our results do not assume equilibrium or aggregating utility functions, and our assumptions about returns concern only the dynamics of the ICML.

The two fund separation theorem allows us subsequently to aggregate over investors and derive a single factor CAPM where the first and second moments of security returns may change stochastically over time and markets are potentially incomplete.

In equilibrium, under the assumptions of our two-fund separation theorem, the market portfolio will be proportional to the logarithmic portfolio. Hence, the single-factor ICAPM holds, even though means, variances, and covariances may change stochastically over time.

The paper is organized as follows. Section 2 outlines the trading model. Section 3 defines the ICML and discusses consumption and portfolio strategies. Section 4 states and proves our two-fund separation theorem, while Section 5 discusses its interpretation and implications. In Section 6, we note that our result implies the instantaneous CAPM in equilibrium. We conclude in Section 7.

2 The model

The notation follows Nielsen...

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